Question

in circle c, what is mfh? 121° 31° 48° 112°

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Find the measure of arc $\overset{\frown}{AJ}$

Let's assume we use the relationship between angles and arcs. If we consider the angles formed by the chords and use the fact that the sum of angles in a triangle and the inscribed - angle property. Let's say we know that if we consider the angles at the circumference related to the arcs. But we need to use the property: If we consider the angle $\angle DEA = 37^{\circ}$, the arc $\overset{\frown}{AB}$ is related to it. And if we consider the angle $\angle GFH=32^{\circ}$, we know that the measure of an inscribed angle $\theta$ and its intercepted arc $s$ satisfy $\theta=\frac{1}{2}s$.
Let's assume we use the fact that the sum of arcs in a circle is $360^{\circ}$. However, a more direct way is to use the property that if we consider the angles subtended by the arcs at the circumference. Let's say we know that the measure of an inscribed angle is half of the measure of the central angle subtended by the same arc.
If we consider the angles formed by the intersecting chords and use the formula for the angle formed by two chords intersecting inside a circle: $\angle AEC=\frac{1}{2}(\overset{\frown}{AC}+\overset{\frown}{BJ})$. But we can also use the fact that if we consider the angles $\angle DEA = 37^{\circ}$ and $\angle GFH = 32^{\circ}$ and the angle $\angle AEC=38^{\circ}$.
We know that the measure of an inscribed angle $\angle G$ with intercepted arc $\overset{\frown}{FH}$ is given by $\angle G=\frac{1}{2}\overset{\frown}{FH}$.
Since $\angle G = 32^{\circ}$, then $\overset{\frown}{FH}=2\times\angle G$.

Step3: Calculate the measure of arc $\overset{\frown}{FH}$

$\overset{\frown}{FH}=2\times32^{\circ}=64^{\circ}$
But this is wrong. Let's use another approach.
We use the property that the angle formed by two secants from an external point.
Let's assume we consider the angles subtended by the arcs at the circumference.
If we consider the angles $\angle D = 37^{\circ}$ and $\angle G=32^{\circ}$ and the angle formed by the intersection of the lines inside the circle.
We know that the measure of the angle formed by two secants from an external point $P$ to a circle with intercepted arcs $m$ and $n$ is $\theta=\frac{1}{2}(m - n)$.
However, a simpler way is to use the inscribed - angle theorem.
The measure of an inscribed angle $\angle G$ whose intercepted arc is $\overset{\frown}{FH}$ gives us:
Since $\angle G = 32^{\circ}$, by the inscribed - angle theorem $\text{m}\overset{\frown}{FH}=2\times\angle G$.
$\text{m}\overset{\frown}{FH} = 64^{\circ}$
There seems to be some missing information in the problem - setup in the original way. But if we assume the basic inscribed - angle relationship for the angle $\angle G$ which intercepts arc $\overset{\frown}{FH}$:

Answer:

None of the given options (64° is not among 121°, 31°, 48°, 112°)

It's possible that there is a mis - understanding of the problem or some key information is not presented clearly in the figure. If we assume that we are using the inscribed - angle formula $\theta=\frac{1}{2}\text{arc}$, where $\theta$ is the inscribed angle and $\text{arc}$ is the intercepted arc, and we know that $\theta = 32^{\circ}$ (angle $\angle G$) intercepting arc $\overset{\frown}{FH}$, then $\text{arc}\overset{\frown}{FH}=2\times32^{\circ}=64^{\circ}$.